theotherthing: Solution
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\subsection{Protecting {\thethings}}
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\label{subsec:lmdk-sel-sol}
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\subsubsection{{\Thething} set options}
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\label{subsec:lmdk-set-opts}
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This step aims to select a set of candidate {\thething} timestamps options either by randomizing the actual timestamps (Section~\ref{subsec:lmdk-rnd}), or by inserting dummy timestamps (Section~\ref{subsec:lmdk-dum-gen}) to the actual {\thething} timestamps.
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\paragraph{{\Thething} randomization}
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\label{subsec:lmdk-rnd}
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A simple way to select a set of timestamps without disclosing the actual {\thethings} is by \emph{randomly} selecting an equally sized set of timestamps.
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The randomization of the process, as we will discuss in more detail in Section~\ref{subsec:priv-opt-sel}, will depend on the positioning of the {\thethings} in the series of events.
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In more detail, given a set of {\thething} timestamps $\{l_k\} \subseteq \{t_n\}$, where $\{t_n\}$ is an event sequence, we need to select all possible sets of size $k$ from $\{t_n\}$.
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However, the introduction of randomization could impact arbitrarily the effectiveness of non-uniform privacy-protection methods.
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This applies mainly in cases where we try to achieve optimal privacy-protection of {\thething} events while maximizing the utility of the data that corresponds to the rest of the series of events.
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As a consequence, it is possible to end up providing lower levels of protection to {\thething} data than the one necessary, i.e.,~worse than the users' privacy-protection expectations.
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The methodology that we present next (Section~\ref{subsec:lmdk-dum-gen}) attempts to tackle the aforementioned shortcoming.
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\paragraph{Dummy {\thething} generation}
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\label{subsec:lmdk-dum-gen}
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Selecting extra events, on top of the actual {\thethings}, as dummy {\thethings} can render actual ones indistinguishable.
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The goal is to select a list of sets with additional timestamps from a series of events at timestamps $\{t_n\}$ for a set of {\thethings} at $\{l_k\} \subseteq \{t_n\}$.
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Algorithms~\ref{algo:lmdk-sel-opt} and \ref{algo:lmdk-sel-heur} approach this problem with an optimal and heuristic methodology, respectively.
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Function \calcMetric measures an indicator for the union of $\{l_k\}$ and a timestamp combination from $\{t_n\} \setminus \{l_k\}$.
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Function \evalSeq evaluates the result of \calcMetric by, e.g.,~estimating the standard deviation of all the distances from the previous/next {\thething}.
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Function \getOpts returns all possible \emph{valid} sets of combinations \opt such that $\{l_{k+i}\} \subset \{l_{k+j}\}, \forall i, j \in [k, n] \mid i < j$, i.e.,~larger options must contain all of the timestamps that are present in smaller ones.
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Each combination contains a set of timestamps with sizes $k + 1, k + 2, \dots, n$, where each one of them is a combination of $\{l_k\}$ with $x \in [1, n - k]$ timestamps from $\{t_n\}$.
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\begin{algorithm}
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\caption{Optimal dummy {\thething} set options selection}
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\label{algo:lmdk-sel-opt}
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\DontPrintSemicolon
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\KwData{$\{t_n\}, \{l_k\}$}
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\SetKwInput{KwData}{Input}
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\KwResult{\optim}
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\BlankLine
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% Evaluate the original
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\metricOrig $\leftarrow$ \calcMetric{$\{t_n\}, \emptyset, \{l_k\}$}\;
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\evalOrig $\leftarrow$ \evalSeq{\metricOrig}\;
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% Get all possible option combinations
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\opts $\leftarrow$ \getOpts{$\{t_n\}, \{l_k\}$}\;
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% Track the minimum (best) evaluation
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\diffMin $\leftarrow$ $\infty$\;
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% Track the optimal sequence (the one with the best evaluation)
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\optim $\leftarrow$ $[]$\;
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\ForEach{\opt $\in$ \opts}{\label{algo:lmdk-sel-opt-for-each}
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\evalSum $\leftarrow 0$\;
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\ForEach{\opti $\in$ \opt}{
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\metricCur $\leftarrow$ \calcMetric{$\{t_n\}, \opti, \{l_k\}$}\;\label{algo:lmdk-sel-opt-comparison}
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\evalSum $\leftarrow$ \evalSum $+$ \evalSeq{\metricCur}\;
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% Compare with current optimal
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\diffCur $\leftarrow \left|\evalSum/\#\opt - \evalOrig\right|$\;
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\If{\diffCur $<$ \diffMin}{
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\diffMin $\leftarrow$ \diffCur\;
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\optim $\leftarrow$ \opt\;
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}
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}
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}\label{algo:lmdk-sel-opt-end}
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\Return{\optim}
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\end{algorithm}
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Algorithm~\ref{algo:lmdk-sel-opt}, in particular, between Lines~{\ref{algo:lmdk-sel-opt-for-each}-\ref{algo:lmdk-sel-opt-end}} evaluates each option in \opts.
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It finds the option that is the most \emph{similar} to the original (Lines~{\ref{algo:lmdk-sel-opt-comparison}-\ref{algo:lmdk-sel-opt-end}}), i.e.,~the option that has an evaluation that differs the least from that of the sequence $\{t_n\}$ with {\thethings} $\{l_k\}$.
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\begin{algorithm}
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\caption{Heuristic dummy {\thething} set options selection}
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\label{algo:lmdk-sel-heur}
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\DontPrintSemicolon
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\KwData{$\{t_n\}, \{l_k\}$}
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\KwResult{\optim}
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\BlankLine
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% Evaluate the original
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\metricOrig $\leftarrow$ \calcMetric{$\{t_n\}, \emptyset, \{l_k\}$}\;
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\evalOrig $\leftarrow$ \evalSeq{\metricOrig}\;
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% Get all possible option combinations
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\optim $\leftarrow$ $[]$\;
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$\{l_{k'}\} \leftarrow \{l_k\}$\;
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\While{$\{l_{k'}\} \neq \{t_n\}$}{\label{algo:lmdk-sel-heur-while}
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% Track the minimum (best) evaluation
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\diffMin $\leftarrow$ $\infty$\;
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\optimi $\leftarrow$ $0$\;
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% Find the combinations for one more point
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\ForEach{\reg $\in \{t_n\} \setminus \{l_{k'}\}$}{
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% Evaluate current
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\metricCur $\leftarrow$ \calcMetric{$\{t_n\}, \reg, \{l_{k'}\}$}\;\label{algo:lmdk-sel-heur-comparison}
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\evalCur $\leftarrow$ \evalSeq{\metricCur}\;
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% Compare evaluations
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\diffCur $\leftarrow$ $\left|\evalCur - \evalOrig\right|$\;
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\If{\diffCur $<$ \diffMin}{
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\diffMin $\leftarrow$ \diffCur\;
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\optimi $\leftarrow$ \reg\;
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}\label{algo:lmdk-sel-heur-comparison-end}
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}
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% Save new point to landmarks
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$k' \leftarrow k' + 1$\;
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$l_{k'} \leftarrow \optimi$\;
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% Add new option
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\optim.add($\{l_{k'}\} \setminus \{l_k\}$)\;
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}\label{algo:lmdk-sel-heur-end}
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\Return{\optim}
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\end{algorithm}
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Algorithm~\ref{algo:lmdk-sel-heur}, follows an incremental methodology.
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At each step it selects a new timestamp that corresponds to a regular ({non-\thething}) event from $\{t_n\} \setminus \{l_k\}$.
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Similar to Algorithm~\ref{algo:lmdk-sel-opt}, the selection is done based on a predefined metric (Lines~{\ref{algo:lmdk-sel-heur-comparison}-\ref{algo:lmdk-sel-heur-comparison-end}}).
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This process (Lines~{\ref{algo:lmdk-sel-heur-while}-\ref{algo:lmdk-sel-heur-end}}) goes on until we select a set that is equal to the size of the series of events, i.e.,~$\{l_{k'}\} = \{t_n\}$.
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Note that the reverse heuristic approach, i.e.,~starting with $\{t_n\}$ {\thethings} and removing until $\{l_k\}$, performs worse than and occasionally the same with Algorithm~\ref{algo:lmdk-sel-heur}.
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\subsubsection{Privacy-preserving option selection}
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\label{subsec:priv-opt-sel}
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% Nearby events
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Events that occur at recent timestamps are more likely to reveal sensitive information regarding the users involved~\cite{kellaris2014differentially}.
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Thus, taking into account more recent events with respect to {\thethings} can result in less privacy loss and better privacy protection overall.
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This leads to worse data utility.
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% Depending on the {\thething} discovery technique
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The values of events near a {\thething} are usually similar to that of the latter.
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Therefore, privacy-preserving mechanisms are likely to approximate their values based on the nearest {\thething} instead of investing extra privacy budget to perturb their actual values; thus, spending less privacy budget.
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Saving privacy budget for releasing perturbed versions of actual event values can bring about better data utility.
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% Distant events
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However, indicating the existence of randomized/dummy {\thethings} nearby actual {\thethings} can increase the adversarial confidence regarding the location of the latter within a series of events.
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Hence, choosing randomized/dummy {\thethings} far from the actual {\thethings} (and thus less relevant) can limit the final privacy loss.
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